Subgroups — #14541816

Order 2176.

350 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763537	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763531	C₂	1	1	yes	maximal, characteristic
3	#11763203	C₂	1	1	yes	maximal, characteristic
4	#11763202	C₂	1	1	yes	maximal, characteristic
5	#11763187	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763034	C₂	1	1	yes	maximal, characteristic
7	#11763031	C₂	1	1	yes	maximal, characteristic
8	#11763019	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763018	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763018	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763013	C₂	1	1	yes	maximal, characteristic
12	#11762937	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762915	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762896	C₂	1	1	yes	maximal, characteristic
15	C₁₇×#441	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
19	#10587633	C₂²	1	1	yes	characteristic
20	#10587630	C₂²	1	1	yes	characteristic
21	#10587630	C₂²	1	1	yes	characteristic
22	#10587630	C₂²	1	1	yes	characteristic
23	C₂×#63231	C₂²	1	1	yes	characteristic
24	C₂×#63231	C₂²	1	1	yes	characteristic
25	C₂×#63231	C₂²	1	1	yes	characteristic
26	C₂×#63230	C₂²	1	1	yes	characteristic
27	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
28	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
29	C₂×#63219	C₂²	1	1	yes	characteristic
30	C₂×#63219	C₂²	1	1	yes	characteristic
31	#10587584	—	2	1	no	—
32	#10587582	—	2	1	no	—
33	#10587579	C₂²	1	1	yes	characteristic
34	#10587575	—	2	1	no	—
35	#10587575	—	2	1	no	—
36	#10587573	C₂²	1	1	yes	characteristic
37	#10587573	C₂²	1	1	yes	characteristic
38	#10587573	C₂²	1	1	yes	characteristic
39	#10587562	—	2	1	no	—
40	#10587561	C₂²	1	1	yes	characteristic
41	#10587558	—	2	1	no	—
42	#10587556	—	2	1	no	—
43	#10587555	C₂²	1	1	yes	characteristic
44	C₄×Dic₃₄	—	2	1	no	—
45	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
46	C₁₇×#106	C₂²	1	1	yes	characteristic
47	C₁₇×#103	C₂²	1	1	yes	characteristic
48	C₁₇×#103	C₂²	1	1	yes	characteristic
49	C₁₇×#97	C₂²	1	1	yes	characteristic
50	#10587486	C₂²	1	1	yes	characteristic
51	#10587485	C₂²	1	1	yes	characteristic
52	#10587485	C₂²	1	1	yes	characteristic
53	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
54	#10587481	C₂²	1	1	yes	characteristic
55	#10587481	C₂²	1	1	yes	characteristic
56	#10587481	C₂²	1	1	yes	characteristic
57	#10587480	C₂²	1	1	yes	characteristic
58	C₄⋊#3622	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
61	C₂×D₆₈	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₁₇	—	2	1	no	—
63	C₂×Dic₃₄	D₄	1	1	yes	characteristic
64	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
65	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
66	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
67	Q₁₆×C₁₇	—	2	2	no	—
68	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₁₇	—	2	1	no	—
71	C₄⋊C₄×C₁₇	—	2	1	no	—
72	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
74	C₄²×C₁₇	—	2	1	no	—
75	C₄²×C₁₇	—	2	1	no	—
76	C₄²×C₁₇	C₂³	1	1	yes	characteristic
77	#63231	—	4	1	no	—
78	#63231	—	4	1	no	—
79	#63231	—	4	1	no	—
80	#63230	—	4	1	no	—
81	#63227	—	2	1	no	—
82	#63227	—	2	1	no	—
83	#63227	—	2	1	no	—
84	#63227	—	2	1	no	—
85	#63227	—	2	1	no	—
86	C₄⋊Dic₁₇	—	2	1	no	—
87	C₄⋊Dic₁₇	—	2	1	no	—
88	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
89	#63225	—	2	1	no	—
90	#63225	—	2	1	no	—
91	#63225	—	2	1	no	—
92	#63225	—	2	1	no	—
93	#63225	—	2	1	no	—
94	#63225	—	2	1	no	—
95	#63225	—	2	1	no	—
96	#63225	—	2	1	no	—
97	#63225	—	2	1	no	—
98	#63225	D₄	1	1	yes	characteristic
99	#63225	D₄	1	1	yes	characteristic
100	C₄×Dic₁₇	—	2	1	no	—
101	C₄×Dic₁₇	—	2	1	no	—
102	C₄×Dic₁₇	—	2	1	no	—
103	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
104	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
105	C₂×#3622	C₂³	1	1	yes	characteristic
106	C₂×#3622	C₂³	1	1	yes	characteristic
107	Dic₆₈	—	2	2	no	—
108	Dic₆₈	—	4	1	no	—
109	#63219	—	2	2	no	—
110	#63219	—	4	1	no	—
111	C₂²×D₁₇	—	2	1	no	—
112	Q₈×C₁₇	—	2	1	no	—
113	Q₈×C₁₇	—	2	1	no	—
114	Q₈×C₁₇	—	2	1	no	—
115	Q₈×C₁₇	—	2	1	no	—
116	C₂×C₄×C₁₇	—	2	1	no	—
117	C₂×C₄×C₁₇	—	2	1	no	—
118	C₂×C₄×C₁₇	—	2	1	no	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
123	C₂×Dic₁₇	—	2	1	no	—
124	C₂×Dic₁₇	—	2	1	no	—
125	C₂×Dic₁₇	—	2	1	no	—
126	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
130	D₆₈	—	2	1	no	—
131	D₆₈	—	2	1	no	—
132	C₄×D₁₇	—	4	2	no	—
133	Dic₃₄	—	2	2	no	—
134	Dic₃₄	—	2	1	no	—
135	Dic₃₄	—	2	1	no	—
136	Dic₃₄	—	2	1	no	—
137	Dic₃₄	—	2	1	no	—
138	Dic₃₄	—	2	1	no	—
139	Dic₃₄	—	2	1	no	—
140	C₈×C₁₇	—	2	1	no	—
141	C₈×C₁₇	#41	1	2	yes	—
142	#3622	—	2	1	no	—
143	#3622	—	2	1	no	—
144	#3398	—	17	1	no	maximal, Sylow
145	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₁₇	—	4	1	no	—
147	C₂×D₁₇	—	4	1	no	—
148	C₂×D₁₇	—	4	1	no	—
149	C₄×C₁₇	—	2	1	no	—
150	C₄×C₁₇	—	2	1	no	—
151	C₄×C₁₇	—	4	1	no	—
152	C₄×C₁₇	—	4	1	no	—
153	C₄×C₁₇	—	4	1	no	—
154	C₄×C₁₇	—	4	1	no	—
155	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
156	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
157	Dic₁₇	—	2	1	no	—
158	Dic₁₇	—	2	1	no	—
159	Dic₁₇	—	2	1	no	—
160	Dic₁₇	—	4	1	no	—
161	Dic₁₇	—	4	1	no	—
162	Dic₁₇	—	4	1	no	—
163	Dic₁₇	#137	1	2	yes	—
164	#557	—	17	1	no	—
165	#556	—	17	1	no	—
166	#497	—	17	1	no	—
167	#497	—	17	1	no	—
168	#494	—	17	1	no	—
169	#464	—	17	1	no	—
170	#464	—	17	1	no	—
171	#462	—	17	1	no	—
172	#462	—	17	1	no	—
173	#462	—	17	1	no	—
174	#461	—	17	1	no	—
175	#446	—	17	1	no	—
176	#441	—	17	1	no	—
177	#441	D₁₇	1	1	yes	characteristic, complemented
178	C₄×SD₁₆	—	17	1	no	—
179	C₂×C₁₇	#550	1	1	yes	characteristic
180	C₂×C₁₇	C₂×#137	1	1	yes	characteristic
181	C₂×C₁₇	#578	1	1	yes	characteristic
182	D₁₇	—	8	1	no	—
183	C₂×Q₁₆	—	17	1	no	—
184	C₂×Q₁₆	—	17	1	no	—
185	C₂×Q₁₆	—	17	1	no	—
186	C₂×Q₁₆	—	17	1	no	—
187	C₂×Q₁₆	—	17	1	no	—
188	C₂×Q₁₆	C₂×D₁₇	1	1	yes	characteristic
189	C₂×SD₁₆	—	17	1	no	—
190	C₂×SD₁₆	—	17	1	no	—
191	C₂×SD₁₆	—	17	1	no	—
192	C₄⋊Q₈	—	17	1	no	—
193	C₄⋊Q₈	—	17	1	no	—
194	C₄⋊Q₈	—	17	1	no	—
195	C₄⋊Q₈	—	17	1	no	—
196	#126	—	34	1	no	—
197	#126	—	34	1	no	—
198	#125	—	34	1	no	—
199	#125	—	34	1	no	—
200	#125	—	34	1	no	—
201	#124	—	17	1	no	—
202	#124	—	17	1	no	—
203	C₂²⋊Q₈	—	34	1	no	—
204	C₄×Q₈	—	17	1	no	—
205	C₄×Q₈	—	17	1	no	—
206	C₄×Q₈	—	17	1	no	—
207	C₄×Q₈	—	17	1	no	—
208	C₄×Q₈	—	34	1	no	—
209	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
210	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
211	C₄×D₄	—	17	1	no	—
212	#117	—	34	1	no	—
213	#106	C₂×D₁₇	1	1	yes	characteristic
214	C₄⋊C₈	—	17	1	no	—
215	C₄⋊C₈	—	17	1	no	—
216	C₄⋊C₈	—	17	1	no	—
217	#103	—	17	1	no	—
218	#103	—	17	1	no	—
219	#103	—	17	1	no	—
220	#103	C₂×D₁₇	1	1	yes	characteristic
221	#103	C₂×D₁₇	1	1	yes	characteristic
222	#102	—	17	1	no	—
223	#97	—	17	1	no	—
224	#97	C₂×D₁₇	1	1	yes	characteristic
225	C₄×C₈	—	17	1	no	—
226	C₁₇	#3398	1	1	yes	characteristic, complemented, Sylow
227	C₂×Q₈	—	17	1	no	—
228	C₂×Q₈	—	17	1	no	—
229	C₂×Q₈	—	17	1	no	—
230	C₂×Q₈	—	17	1	no	—
231	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
232	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
233	C₂×D₄	—	17	1	no	—
234	C₂²×C₄	—	34	1	no	—
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	34	2	no	—
237	Q₁₆	—	68	1	no	—
238	Q₁₆	—	68	1	no	—
239	Q₁₆	—	68	1	no	—
240	Q₁₆	—	68	1	no	—
241	SD₁₆	—	34	2	no	—
242	SD₁₆	—	68	1	no	—
243	SD₁₆	—	68	1	no	—
244	C₂×C₈	—	17	1	no	—
245	C₂×C₈	—	17	1	no	—
246	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	17	1	no	—
251	C₄⋊C₄	—	17	1	no	—
252	C₄⋊C₄	—	17	1	no	—
253	C₄⋊C₄	—	34	1	no	—
254	C₄⋊C₄	—	34	1	no	—
255	C₄⋊C₄	—	34	1	no	—
256	C₄⋊C₄	—	34	1	no	—
257	C₄⋊C₄	—	34	1	no	—
258	C₄⋊C₄	—	34	1	no	—
259	C₄⋊C₄	—	34	1	no	—
260	C₄⋊C₄	—	34	1	no	—
261	C₄⋊C₄	—	34	1	no	—
262	C₄⋊C₄	—	34	1	no	—
263	C₄⋊C₄	—	34	1	no	—
264	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
266	C₂²⋊C₄	—	34	1	no	—
267	C₂²⋊C₄	—	34	1	no	—
268	C₂²⋊C₄	—	34	1	no	—
269	C₂²⋊C₄	—	34	1	no	—
270	C₂²⋊C₄	—	34	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	17	1	no	—
274	C₄²	—	17	1	no	—
275	C₄²	—	34	1	no	—
276	C₄²	—	34	1	no	—
277	C₄²	—	34	1	no	—
278	C₄²	C₂²×D₁₇	1	1	yes	characteristic
279	C₂³	—	34	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	34	2	no	—
285	Q₈	—	34	1	no	—
286	Q₈	—	34	1	no	—
287	Q₈	—	34	1	no	—
288	Q₈	—	34	1	no	—
289	Q₈	—	34	1	no	—
290	Q₈	—	34	1	no	—
291	D₄	—	34	1	no	—
292	D₄	—	34	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	17	1	no	—
298	C₂×C₄	—	17	1	no	—
299	C₂×C₄	—	17	1	no	—
300	C₂×C₄	—	17	1	no	—
301	C₂×C₄	—	34	1	no	—
302	C₂×C₄	—	34	1	no	—
303	C₂×C₄	—	34	1	no	—
304	C₂×C₄	—	68	2	no	—
305	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
306	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	34	1	no	—
310	C₈	—	34	1	no	—
311	C₈	#63252	1	2	yes	—
312	C₂²	—	68	1	no	—
313	C₂²	—	68	1	no	—
314	C₂²	—	68	1	no	—
315	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	17	2	no	—
323	C₄	—	34	1	no	—
324	C₄	—	34	1	no	—
325	C₄	—	34	1	no	—
326	C₄	—	68	1	no	—
327	C₄	—	68	1	no	—
328	C₄	—	68	1	no	—
329	C₄	C₂×#63252	1	1	yes	characteristic
330	C₄	#10587696	1	1	yes	characteristic
331	C₂	—	136	1	no	—
332	C₂	#11763480	1	1	yes	characteristic
333	C₂	#137×D₁₇	1	1	yes	characteristic
334	C₂	#11763625	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#441	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	C₂×D₆₈	C₂³
hypercenter	#441	D₁₇
hypocenter	C₁₇	#3398
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #11763537	C₂	semidirect
5. #11763187	C₂	semidirect
8. #11763019	C₂	semidirect
9. #11763018	C₂	semidirect
10. #11763018	C₂	semidirect
12. #11762937	C₂	semidirect
13. #11762915	C₂	semidirect
15. C₁₇×#441	C₂	semidirect
177. #441	D₁₇	semidirect
226. C₁₇	#3398	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#11763537	2
#11763531	2
#11763203	2
#11763202	2
#11763187	2
#11763034	2
#11763031	2
#11763019	2
#11763018	2
#11763013	2
#11762937	2
#11762915	2
#11762896	2
C₁₇×#441	2
#3398	17